I will present a general framework for solving partial differential equations on manifolds represented by meshless points, i.e., point clouds, without parametrization or connection information. Our method is based on a local approximation of the manifold, such as using least squares, in a local intrinsic coordinate system constructed by local principal component analysis (PCA) using K-nearest neighbors (KNN). Once the local reconstruction is available, differential operators on the manifold can be approximated discretely.
The framework extends to manifolds of any dimension. The complexity of our method scales well with the total number of points and the true dimension of the manifold (not the embedded dimension). Different least square approximations, treatment of boundary conditions, approximation error analysis, and numerical tests will be presented.
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